1280-ch21 p 355

21 Acceptable Points in Games of Perfect Information

Summary

This is the second of a series of papers on the theory of acceptable points

in n-person games. The first was [1]; in it the notion of acceptable point

was defined for cooperative games, and a fundamental theorem was

proved relating the acceptable expected payo¤s for a single play of a

game to probable average payo¤s for ‘‘strong equilibrium points’’ in its

supergame.1

The chief result of the current paper, Theorem 5.4, is a generalization

of von Neumann’s classical theorem on two-person zero-sum games of

perfect information (see [11]). Roughly, it states that strong equilibrium

points in the supergame of a stable game of perfect information can be

achieved in pure supergame strategies. An example shows that not all

games possess this property; and in fact, it is conjectured that the prop-

erty is characteristic of game structures of perfect information.

The theorem stated above holds whether G is interpreted as a coopera-

tive or as a non-cooperative game. To lend meaning to this statement,

we will have to extend the theory introduced in [1] to non-cooperative

games. We plan to do this in full in a subsequent paper. Here just enough

definitions and theorems will be used to enable us to state and prove the

chief result for non-cooperative games of perfect information.

The paper is divided into two parts, the first centering around the proof

of the chief result for cooperative games, the second dealing with the

extension to non-cooperative games. Section 1, the introduction, serves

mainly to supply background from [1] and from the literature. In O2, we

show that the naive approach to generalizing von Neumann’s theorem

on games of perfect information fails; that is, we bring an example of a

stable game of perfect information that has no acceptable point in pure

strategies. It is then shown intuitively that an appropriate generalization

of the von Neumann Theorem should involve the supergame. Sections 3

and 4 are devoted to the proof of preliminary theorems, dealing with

supergame pure strategies and supergames of perfect information,

respectively. In O5 we establish the chief result. Section 6, which com-

pletes the first part of the paper, is devoted to the example and conjecture

mentioned in connection with the chief result.

The second part begins with O7, a summary of the additional notation

needed for the non-cooperative case. In O8 the concept of acceptability

This chapter originally appeared in Pacific Journal of Mathematics 10 (1960): 381–417.Reprinted with permission.

1. Readers not familiar with [1] should read the introduction (section 1) before continuingwith this summary.

is defined for non-cooperative games. In O9, we show that in a game G

of perfect information, it makes no di¤erence, insofar as the theory of

acceptable points is concerned, whether G is to be considered as a non-

cooperative or as a cooperative game. More precisely, it is shown that the

set of acceptable payo¤s in the non-cooperative sense, coincides with the

set of acceptable payo¤s in the cooperative sense. This is a consequence

of the lemma, interesting in its own right, that in a game of perfect infor-

mation, the set of payo¤ vectors to correlated strategy vectors coincides

with the set of payo¤ vectors to mixed strategy vectors. Again, this

lemma seems to be characteristic of game structures of perfect informa-

tion. In O10, we define supergame strategies for non-cooperative games

and prove some preliminary results. Section 11 is devoted to the state-

ment and proof of the chief result for non-cooperative games of perfect

information.

As in [1], the games under consideration contain no chance moves.

We will make unrestricted use of the notations, ideas, definitions, the-

orems and proofs of [1]. We will not in general repeat explanations and

proofs that are similar to those given there. Especially heavy use will be

made of O6 of [1].

1 Introduction and Background

Up to the present, the starting point for all work on games of perfect

information has been the theorem of von Neumann that every two-

person zero-sum game of perfect information with finitely many moves

has a solution in pure strategies. Subsequent work has dealt with exten-

sions to n-person games and the concomitant generalizations of the solu-

tion notion, with various converses to the von Neumann theorem, with

extensions to games containing infinitely many moves (i.e., positions),

and with various combinations of these. We mention also the notion of

stochastic games of perfect information with infinitely many moves.

In the first of these areas, Kuhn [9] showed that the von Neumann

theorem could be extended to n-person games if the ‘‘equilibrium point’’

notion of Nash [12] was substituted for the classical solution notion.

Dalkey [4] proved a converse of this theorem, which reduces to a con-

verse of the von Neumann theorem in the two-person, zero-sum case.

Gale and Stewart [6] were the first to treat games of perfect information

with infinitely many moves; they showed that certain such (two-person

zero-sum) games possess no pure strategy solutions, and derived su‰cient

conditions for the existence of a pure strategy solution. Wolfe [14]

extended their results. By adopting a definition of payo¤ that is some-

Strategic Games: Repeated356

what more restricted than that of Gale and Stewart, Berge [2] was able

to extend von Neumann’s theorem to some games with infinitely many

moves. He was also able to show [2, 3] that under very general conditions

on the structure of the game, Kuhn’s theorem on the existence of a pure

strategy equilibrium point in a game of perfect information holds true.

The work of Shapley [13] and Gillette [7] on Stochastic Games of perfect

information will be discussed in detail below.

The current paper deals with an extension of the von Neumann

theorem to n-person games. The solution notion that we use is that of

‘‘acceptable’’ points, introduced in [1]. The notion of acceptability is a

generalization of the ‘‘core’’ introduced by Gillies [8] for the cooperative

game with side payments. More precisely, an n-tuple x of strategies is

called acceptable if the players of any given coalition can be prevented by

the players not in that coalition from each obtaining a higher payo¤ than

when x is played (Definition 4.1 of [1]). Intuitively, it would seem that in

a long sequence of plays of a game, a ‘‘steady state’’ would have to rep-

resent an acceptable point, because the players would certainly tend to

move away from any point that is not acceptable.

In order to obtain a precise statement and proof of this intuitive idea,

we introduced (in O6 of [1]) the formal notion of the ‘‘supergame’’ of a

given game G. The supergame of G is a game each play of which consists

of an infinite sequence of plays of G. The payo¤ to a superplay (i.e., a

play of the supergame) is given by the average (i.e., first cesaro limit, if it

exists) of the payo¤s to the individual plays of G that constitute the

superplay. Many of the notions that apply to ordinary games can also be

applied to supergames. In particular, it is possible to define the notion of

strategy in the supergame, and also the notion of a strategy equilibrium

point in the sense of Nash. A much stronger form of the Nash equilib-

rium notion may be defined as follows: An n-tuple x of strategies is called

a ‘‘strong equilibrium point’’ if for no coalition B can all the members of

B increase their payo¤ by adopting strategies di¤erent from those at x

while the remaining players (those in N�B) play as they did at x. The

notion of strong equilibrium applied to the supergame provides a for-

malization of the ‘‘steady state’’ idea (O7 of [1]).

The basic result of [1] (O10) may be stated as follows: The payo¤s for

the acceptable points in a game G are the same as the payo¤s for the

strong equilibrium points in its supergame. Since the notion of accept-

ability depends only on the payo¤, this means that the acceptable points

in G correspond precisely to the steady state points in the supergame of

G. For two-person zero-sum games, a point is acceptable if and only if its

payo¤ is the game value, whereas it is a strong equilibrium point if and

only if it is a solution (O5 of [1]).

Acceptable Points in Games of Perfect Information357

The object of this paper is to apply the theory of acceptable points

to games of perfect information, with a view to obtaining an appropriate

n-person generalization of the von Neumann theorem. In other words, we

want to accomplish for acceptable points in games of perfect information

what Kuhn did in [9] for equilibrium points in games of perfect informa-

tion. The first conjecture in this direction might be that every game of

perfect information has an acceptable point in pure strategies. This is

unreasonable, because according to an example given in [1] (O11), not

every game of perfect information need have an acceptable point at all,

let alone one in pure strategies. However, it turns out that not even all

stable games (games that do have acceptable points) of perfect informa-

tion have pure strategy acceptable points. The reasons for this are dis-

cussed in O2, and it is also shown there that a more appropriate place to

look for a generalization of the von Neumann theorem is in the super-

game. We would like to show that if G is a game of perfect information,

then each player can restrict himself to pure strategies in each play of

an infinite sequence of plays of G. In fact, we prove (Theorem 5.4) that

every acceptable point (and hence every strong equilibrium point) in a

game of perfect information can be ‘‘achieved’’ in pure supergame strat-

egies, in the sense that there is a pure strategy strong equilibrium point

with the same payo¤. In particular, if the supergame of a game of perfect

information has a strong equilibrium point at all, then it already has one

in pure strategies.

Formally, the supergame defined in [1] bears some resemblance to the

stochastic games treated by Gillette in [7]. The two concepts are similar in

that both involve games consisting of an infinite sequence of plays of

finite games, and the payo¤s in both cases are given by a form of the

average of the payo¤s to the individual plays. The main di¤erences are

that Gillette considers a set of M games, any one of which may be the

game played at a given stage, whereas we are concerned with repeated

plays of one game only; and that Gillette considers two-person zero-sum

games, while we deal with n-person games. The ‘‘intersection’’ of the two

theories is an infinite sequence of plays of the same two-person zero-sum

game of perfect information, a trivial situation once von Neumann’s the-

orem is known (obviously both players play their optimal pure strategies

on each play). The two theories provide totally ‘‘disjoint’’ generalizations

of the von Neumann theorem.

All of Gillette’s positive results involve ‘‘stationary’’ strategies, i.e.,

supergame strategies that are obtained by repeating the same strategy on

each play of the infinite sequence of plays that constitutes a superplay. In

a somewhat similar situation, Everett [5] gives a formal definition of

some strategies that are not stationary, and obtains positive results with


them; but the strategies he defines are still ‘‘almost’’ stationary in the

sense that the choice of a player at a given game of the supergame can

depend only on which game he is at, not on the choices of the other

players on previous occasions.

It is of interest to ask whether these restricted notions of strategy are

su‰cient for our theory. The answer is no. The existence of a strong

equilibrium point in stationary pure strategies would imply the existence

of an acceptable point in pure strategies; and the example in O2 shows

that even in stable games of perfect information such an acceptable

point in pure strategies need not always exist. The same example shows

that there is no strong equilibrium point in ‘‘almost’’ stationary pure

strategies.

Finally, we mention that in the supergames of games of perfect infor-

mation (even unstable ones), there is always a Nash equilibrium point (as

opposed to a strong equilibrium point) in stationary pure strategies. This

is a consequence of Kuhn’s theorem.

2 Failure of the Naive Approach

We saw in [1] (O5) that the concept of acceptability constitutes a general-

ization of the concept of solution in two-person, zero-sum games. As

a generalization of von Neumann’s theorem on two-person zero-sum

games of perfect information, we might hope that every game of perfect

information that has any acceptable points also has acceptable points in

pure strategies. An example shows that this is false.

The game G is a two-person, non-zero-sum game of perfect informa-

tion. In the game tree, given in figure 1, the moves are labelled with the

names of the players and the terminals with the payo¤ vectors. Each

player has two strategies, the left and the right strategies. Notation in the

following payo¤ matrix is obvious.

L1 R1

L2 (6, 0) (2, 1)

R2 (0, 6) (2, 1)

Player 1 cannot be prevented from obtaining at least 2 (he can play R1);

player 2 cannot be prevented from obtaining at least 1 (he can play R2).

This shows that (L1;L2) and (L1;R2) are not acceptable. The other two

pure strategy pairs are not acceptable because the coalition (1, 2) cannot

be prevented from obtaining (3, 3)—by playing (L1; 1=2L2 þ 1=2R2)—

and (3, 3) is strictly larger than the payo¤ vector at both (R1;L2) and

(R1;R2). Hence G has no acceptable point in pure strategies. (Note that


(3, 3) is an acceptable payo¤ vector, so that G does have some acceptable

points.)

The intuitive feeling that a game of perfect information should have a

‘‘good’’ point in pure strategies can be traced to the traditional purpose

of mixed strategies—namely, to hide one’s intentions from one’s oppo-

nent by the use of a random device. In a game of perfect information, we

somehow feel that it is unnecessary to hide one’s intentions, that it is in

the nature of the game that everything may just as well be open and

above-board. The conclusion is that mixed strategies are unnecessary in

such a game, and that therefore we may just as well confine ourselves to

the consideration of pure strategies.

The counterexample points up the fallacy in this intuitive argument. It

is quite true that the hiding of one’s intentions, and the concommitant use

of a random device should be unnecessary in a game of perfect infor-

mation. This does not mean, though, that one can achieve one’s desires

by means of pure strategies. Indeed, if there were some means of

mixing one’s strategies other than by the use of a random device, this

would be perfectly satisfactory in Example 3. For example, the pair

(L1; 1=2L2 þ 1=2R2) happens to be acceptable. If, instead of tossing a

coin before each play of a sequence of plays, 2 were to announce before-

hand that he will alternate L2 and R2, this would in no way a¤ect the

actions of 1. Contrary to the situation in, say, penny matching, the pur-

pose of playing 1=2L1 þ 1=2R1 here is simply to achieve a payo¤ not

provided in the matrix, not to avoid ‘‘discovery’’ by the opponent.

This discussion shows that though we cannot expect pure-strategy

acceptable points in a game G of perfect information, we should be able

to expect that the players may, without loss, restrict themselves to pure

strategies in each of the plays that constitute a superplay of G. This is in

fact the case, as we shall see in the sequel.

3 Supergame Pure Strategies

A supergame pure strategy vector (or p-strategy vector) is a c-strategy

vector in which there are no coalitions and the players choose pure strat-

Figure 1


egies on each play. Here the second condition is the essential one; the first

condition is adopted only for convenience. If the first condition were

eliminated, the resulting supergame strategy vectors would be essentially

equivalent to those obtained under our definition.

The formal definition is as follows:

definition 3.1 A supergame c-strategy f i is said to be ‘‘pure’’ if

f ikðyÞ A Pi

for each kX 0 and y A Ji1 � � � � � Ji

k.

We also say that f i is a supergame p-strategy.

The following are lemmas that will be needed later.

lemma 3.2 If f is a supergame p-strategy vector and B is a (possibly

empty) subset of N, then for each kX 0 and y A J1 � � � � � Jk, we have

f N�Bk ðyÞjRN�B ¼ dN�B

e :

Furthermore, for each kX 1, we have

zkð f Þ ¼ ðx1ð f Þ; . . . ; xkð f ÞÞ:

Proof The first statement follows at once from 3.1. The second state-

ment follows by induction from 6.3, 6.4, and 6.5 of [1], and from 3.1.

lemma 3.3 Let f be a supergame p-strategy vector, and let g be a super-

game c-strategy vector for which

gN�B ¼ f N�B:

Let v ¼ ðv1; . . . ; vk; . . .Þ A J1 � � � � � Jk � � � � occur with positive proba-

bility when g is played (see Definition 10.22 of [1]). Then for all kX 1, we

have

vkjRN�B ¼ dN�Be

and for all kX 0, we have

f N�Bk ðvi; . . . ; vkÞ ¼ f N�B

k ððv1jU1; deÞ; . . . ; ðvkjUk; deÞÞ:

Proof The first statement is an immediate consequence of the previous

lemma. As for the second statement, it follows from the first statement

that

f N�Bk ðv1; . . . ; vkÞ ¼ f N�B

k ððvB1 ; vN�B1 Þ; . . . ; ðvBk ; vN�B

k ÞÞ

¼ f N�Bk ððvB1 ; ðvN�B

1 jUN�B1 ; dN�B

e ÞÞ; . . . ; ðvBk ; ðvN�Bk jUN�B

k ; dN�Be ÞÞÞ:


But by Definition 6.1 of [1], f N�Bk is independent of (vB1 ; . . . ; v

Bk ). The

result follows at once.

For a supergame c-strategy vector f, define Skð f Þð¼ SkÞ by

Skð f Þ ¼1

k

Xkj¼1

Hjð f Þ: ð3:4Þ

Parallel to the definition of strong equilibrium c-point, we may make

the following definition:

definition 3.5 A strong equilibrium p-point f is a summable supergame

p-strategy vector for which there is no BHN for which there is a super-

game p-strategy vector g satisfying

gN�B ¼ f N�B ð3:6Þ

and

lim supk!y

mini A B

ðSikðgÞ �Hið f ÞÞ > 0: ð3:7Þ

The set of all strong equilibrium p-points is denoted by Sp. The Con-

dition 3.7 may also be replaced by the following condition:

lim infk!y

ðSBk ðgÞ �HBð f ÞÞ > 0: ð3:8Þ

We denote by ~SSp the set of supergame p-strategy vectors that satisfy a

condition that di¤ers from 3.5 only in that 3.7 is replaced by 3.8.

The essential di¤erence between a strong equilibrium p-point and a

pure strong equilibrium c-point is that in the former, N � B need only be

prepared to defend against all supergame pure strategy B-vectors,

whereas in the latter, N � B must be prepared to defend against all

supergame correlated strategy B-vectors. We will show in 3.11 that the

two conditions are nevertheless equivalent. As for 3.7 and 3.8, they are

merely translations of 7.2 and 7.3 of [1] to the case of pure strategies,

where the consideration of probabilities becomes superfluous.

theorem 3.9 If f is a supergame p-strategy vector, then zkð f Þ, xkð f Þ, andEkð f Þ are ‘‘pure’’ for each kX 0; that is, they are discrete probability dis-

tributions in which one of the events occurs with probability 1, all others

with probability 0.

Proof This is a trivial consequence of (6.2), (6.3), (6.4), (6.5) and (6.6) of

[1], and of 3.1.

Theorem 3.9 enables us to replace probability statements involving the

random variable SkðvÞ by statements involving the constants Skð f Þ only.More precisely, we have


corollary 3.10 Let F(x1; x2; . . .) be a predicate depending on a sequence

of B-vectors x1; x2; . . . : Let L be the propositional function that assigns the

number 1 to true propositions and the number 0 to false propositions. Sup-

pose f is a supergame p-strategy vector for the game G. Then

Probf FðSB1 ðvÞ;SB

2 ðvÞ; . . .Þ ¼ LðFðSB1 ð f Þ;SB

2 ð f Þ � � �ÞÞ

Similar results hold for zkð f Þ; xkð f Þ and Ekð f Þ.

theorem 3.11 Every strong equilibrium p-point is a strong equilibrium

c-point. Conversely, every pure strong equilibrium c-point is a strong equi-

librium p-point. In symbols

Fp XSc ¼ Sp;

where Fp is the set of supergame p-strategy vectors.

Proof We consider first the converse, the easier of the two statements.

Let f be a pure strong equilibrium c-point. It is su‰cient to prove that

there is no pure g satisfying 3.6 and 3.7. Suppose there is such a g. Then g

must satisfy 7.1 of [1], which is identical with 3.6. Furthermore, from 3.7

we deduce the existence of an e > 0 such that for infinitely many k, we

have

mini A B

ðSikðgÞ �Hið f ÞÞ > e:

It follows that for infinitely many k, we have

SBk ðgÞ > HBð f Þ þ eB;

where eB is a B-vector defined by

ei ¼ e

for all i A B. Hence it follows that for all k, we have

SBr ðgÞ > HBð f Þ þ eB for some rX k:

Applying 3.10, we obtain

ProbgðSBr ðvÞXHBð f Þ þ eB for some rX kÞ ¼ 1

for all k. Hence it follows that

limk A y

probgðSBr ðvÞXHBð f Þ þ eB for some rX kÞ ¼ 1 > 0:

But this is exactly Condition 7.2 of [1]. We have established that g sat-

isfies 7.1 and 7.2 of [1], whence f cannot be a strong equilibrium c-point.

This contradicts the hypothesis, and we must conclude that g satisfies 3.6

and 3.7. This completes the proof of the converse.


Now assume that f is a strong equilibrium p-point, but not a strong

equilibrium c-point. Then there is a supergame c-strategy vector g sat-

isfying 7.1 and 7.2 of [1]. From 7.1 of [1] we obtain

gN�B ¼ f N�B: ð1Þ

From 7.2 of [1], we obtain that there is a B-vector eB > 0 for which

limk!y

ProbgðSBr ðvÞXHBð f Þ þ eB for some rX kÞ > 0: ð2Þ

Now the expression inside the limit on the left side of (2) is monotone

decreasing with k; hence (2) implies the existence of a

d > 0 ð3Þ

such that

ProbgðSBr ðvÞXHBð f Þ þ eB for some rX kÞ > d; for all kX 1: ð4Þ

From (4) we obtain

ProbgðFor all kX 1;SBr ðvÞXHBð f Þ þ eB for some rX kÞX d; ð5Þ

which is the same as

ProbgðSBr ðvÞXHBð f Þ þ eB for infinitely many rÞX d: ð6Þ

That (5) follows from (4) is an immediate consequence of the fact that the

measure of the intersection of a monotone decreasing sequence of mea-

surable sets is the limit (or g.l.b.) of the measures of the sets.

From (3) and (6) it follows that there is a sequence

v ¼ ðv1; . . . ; vk; . . .Þ A J1 � � � � � Jk � � � � ;

occurring with positive probability when g is played, for which

SBr ðvÞXHBð f Þ þ eB for infinitely many r: ð7Þ

Since v occurs with positive probability we deduce from 6.4 and 6.2 of

[1], and from the definitions in O2 of [1] that for each k,

0 < sðgk�1ðv1; . . . ; vk�1ÞÞðvkÞW uðcðgk�1ðv1; . . . ; vk�1ÞÞÞðvkjUkÞ

¼X

pk A u�1ðvk jUkÞcðgk�1ðv1; . . . ; vk�1ÞÞð pkÞ:

It follows that for each k, there is a pk satisfying

vkjUk ¼ uðpkÞ ð8Þ

and


cðgk�1ðv1; . . . ; vk�1ÞÞðpkÞ > 0: ð9Þ

Now as a consequence of 2.7 of [1], Lemmas 3.2 and 3.3, and (1), we

have that for each kX 0,

cN�Bðgkðv1; . . . ; vkÞÞ ¼ gN�Bk ðv1; . . . ; vkÞ: ð10Þ

From (9) it follows that

cN�Bðgkðv1; . . . ; vkÞÞðpN�Bkþ1 Þ > 0:

Applying (10), we deduce that

gN�Bk ðv1; . . . ; vkÞðpN�B

kþ1 Þ > 0;

and it then follows from (1) that

f N�Bk ðv1; . . . ; vkÞðpN�B

kþ1 Þ > 0: ð11Þ

Since f N�Bk must be a pure strategy (N � B)-vector, it follows from (11)

that

f N�Bk ðv1; . . . ; vkÞ ¼ pN�B

kþ1 : ð12Þ

We now define a supergame p-strategy vector q by

qN�B ¼ f N�B ð13Þ

qik�1 ¼ pik; i A B; kX 1: ð14Þ

Next, we prove that for kX 1,

zkðqÞ ¼ ððv1jU1; deÞ; . . . ; ðvkjUk; deÞÞ: ð15Þ

That

zkðqÞjR1 � � � � � Rk ¼ ðde; . . . ; deÞ ð16Þ

follows at once from (13), (14) and the fact that f is a supergame p-

strategy vector. The remainder of (15) is proved by induction on k. For

k ¼ 1, we have by 6.2 and 6.3 of [1],

z1ðqÞjU1 ¼ uðcðq0ÞÞ

¼ uðcðpB1 ; f N�B0 ÞÞðby ð13Þ and ð14ÞÞ

¼ uðcðp1ÞÞðby ð12ÞÞ

¼ uðp1Þðby 2:7 of ½1�Þ

¼ v1jU1:

ð17Þ

Now let us assume that we have established


zkðqÞjU1 � � � � �Uk ¼ ðv1jU1; . . . ; vkjUkÞ: ð18Þ

Then by 6.2 and 6.4 of [1],

zkþ1ðqÞjU1 � � � �Ukþ1

¼X�

y A J1��Jk

zkðqÞðyÞðyjU1 � � � � �Uk; uðcðqkðyÞÞÞÞ:

By (16) and (18), all the coe‰cients zkðqÞðyÞ in this sum vanish, unless

y ¼ ððv1jU1; deÞ; . . . ; ðvkjUk; deÞÞ:

Hence

zkþ1ðqÞjU1 � � � � �Ukþ1 ¼ ðv1jU1; . . . ; vkjUk; uðcðqkðzkðqÞÞÞÞÞ: ð19Þ

Now by (14),

qBk ðzkðqÞÞ ¼ pBkþ1 ð20Þ

and by (10),

qN�Bk ðzkðqÞÞ ¼ f N�B

k ðzkðqÞÞ

¼ f N�Bk ððv1jU1; deÞ; . . . ; ðvkjUk; deÞÞðby ð16Þ and ð18ÞÞ

¼ f N�Bk ðv1; . . . ; vkÞ ðby Lemma 3:3Þ

¼ pN�Bkþ1 ðby ð12ÞÞ: ð21Þ

Combining (20) and (21), we obtain qkðzkðqÞÞ ¼ pkþ1.

Hence

cðqkðzkðqÞÞÞ ¼ pkþ1;

and it follows from this and (8) that

uðcðqkðzkðqÞÞÞÞ ¼ uðpkþ1Þ ¼ vkþ1jUkþ1; ð22Þ

Combining (19) and (22), we obtain

zkþ1ðqÞjU1 � � � � �Ukþ1 ¼ ðv1jU1; . . . ; vkþ1jUkþ1Þ:

which completes the inductive step and the proof of (15). Hence (22)

holds for all k, and therefore

Hkþ1ðqÞ ¼ HðcðqkðzkðqÞÞÞÞ ðby 6:6 and 6:7 of ½1�Þ

¼ ðc � uÞðcðqkðzkðqÞÞÞÞ ðby O6 of ½1�Þ

¼ cðuðcðqkðzkðqÞÞÞÞÞ


¼ cðvkþ1jUkþ1Þ ðby ð22ÞÞ

¼ Hkþ1ðvÞ ðby 6:10 of ½1�Þ:

It then follows from 6.11 of [1] and from 3.4 that

SkðqÞ ¼ SkðvÞ:

Applying (7), we obtain that

SBk ðqÞXHBð f Þ þ eB for infinitely many k:

In particular,

mini A B

ðSikðqÞ �Hið f ÞÞX min

i A Bei

for infinitely many k, and it follows that

lim sup mink!y i A B

ðSikðqÞ �Hið f ÞÞX min

i A Bei > 0: ð23Þ

Now by (13) and (14), q is a supergame p-strategy vector. By (1) it sat-

isfies 3.6 and by (23) it satisfies 3.7. Hence by 3.5, f cannot be a strong

equilibrium p-point, a contradiction. This completes the proof of 3.11.

theorem 3.12 Fp X ~SSc ¼ ~SSp.

Proof The proof is similar to that of 3.11. It will be omitted.

The following formulae follow easily from the indicated definitions and

theorems.

HðSpÞHHðScÞ ðby 3:11Þ: ð3:13Þ

Hð ~SSpÞHHð ~SScÞ ðby 3:12Þ: ð3:14Þ

Sp H ~SSp ðby 3:5Þ: ð3:15Þ

HðSpÞHHð ~SSpÞ ðby 3:15Þ: ð3:16Þ

Finally, we mention the following theorem, which will be needed in the

sequel.

theorem 3.17 A supergame p-strategy vector f is summable if and only if

it is summable in the mean.

Proof The necessity follows at once from 6.9 of [1]. For su‰ciency, we

must show that if f is summable in the mean, then a sequence of random

variables distributed according to Ekð f Þ obeys the strong law of large

numbers. But this follows at once from 3.9.


4 Supergame Pure Strategies in Games of Perfect Information

In a game G of perfect information, the information that a player i has

about the outcome of each previous play2 may be described as follows:

He knows which terminal was reached: ð4:1Þ

He knows which pure strategy he himself played: ð4:2Þ

Formally, let W be the set of terminals in G, and let

l : P ! W

be the function that associates with each pure strategy vector p the ter-

minal lðpÞ that results when p is played (in the notation of [9], if

p A P; lðpÞ is the unique w A W for which ppðwÞ ¼ 1). Then for each i A B

and p A P,

uiðpÞ ¼ ðlðpÞ; piÞ: ð4:3Þ

If he wishes, the reader may regard 4.3 as the definition of ui for games of

perfect information.

Actually, each player may with impunity discard the additional infor-

mation obtained from 4.2 as long as he restricts himself to the use of

supergame p-strategies. Formally, we may say that in a game of perfect

information, each supergame p-strategy f i is equivalent to one in which f ikdepends only on the lðpÞ, not on the pi. To lend meaning to this state-

ment, we must give a suitable definition of equivalence.

definition 4.4 Two supergame p-strategies f i and gi are said to be

equivalent ( f i F gi) if for each supergame p-strategy (N � i)-vector yN�i,

we have

Hkð f i; yN�iÞ ¼ Hkðgi; yN�iÞ; kX 1:

corollary 4.5 Let BHN. If two supergame p-strategy vectors f and g

are equivalent, then for each supergame p-strategy (N � B)-vector yN�B,

we have

Hkð f B; yN�BÞ ¼ HkðgB; yN�BÞ; kX 1:

Proof Let

B ¼ fi1; . . . ; ibg:

Then since f i F gi for each i it follows that for each kX 1,

2. We are discussing that information that is characterized by the information function ui .


Hkð f B; yN�BÞ ¼ Hkðgi1 ; ð f B�i1 ; yN�BÞÞ

¼ Hkðgi1 ; gi2 ; ð f B�i1�i2 ; yN�BÞÞ

¼ Hkððgi1 ; . . . ; gibÞ; yN�BÞ

¼ HkðgB; yN�BÞ:

This completes the proof.

definition 4.6 Let G be a game of perfect information. A supergame

p�-strategy f i is a supergame p-strategy for which for each kX 1, and

pair ( p1; . . . ; pk) and (q1; . . . ; qk) of sequences of pure strategy vectors, we

have

lðpjÞ ¼ lðqjÞ; 1W jW k;

) f ikðððlðp1Þ; pi1Þ; deÞ; . . . ; ððlðpkÞ; pikÞ; deÞÞ

¼ f ikðððlðq1Þ; qi1Þ; deÞ; . . . ; ððlðqkÞ; qikÞ; deÞÞ:

For convenience, we will sometimes make use of the following con-

ventions:

convention 4.7 When f i is a supergame p�-strategy, write

f ikðlðp1Þ; . . . ; lðpkÞÞ

instead of

f ikðððlðp1Þ; pi1Þ; deÞ; . . . ; ððlðpkÞ; pikÞ; deÞÞ:

convention 4.8 When f i is a supergame p-strategy, write

f ikðuiðp1Þ; . . . ; uiðpkÞÞ

instead of

f ikððuiðp1Þ; deÞ; . . . ; ðuiðpkÞ; deÞÞ:

The use of these conventions is justified by Definition 4.6 and Lemma

3.3 respectively.

theorem 4.9 In a game G of perfect information, every supergame p-

strategy is equivalent to a supergame p�-strategy.

Proof Let f i be a supergame p-strategy in G. For any sequence of

terminals (a1; . . . ; ak) we may define gikða1; . . . ; ak) by means of the fol-

lowing recursion:

gi0 ¼ f i0 ð1Þ


gijða1; . . . ; ajÞ ¼ f ij ðða1; g10Þ; ða2; gi1ða1ÞÞ; . . . ; ðaj; gij�1ða1; . . . ; aj�1ÞÞÞ;

jW k: ð2Þ

Let yN�i be an arbitrary supergame p-strategy (N � i )-vector. We

prove by induction on k that

zkð f i; yN�iÞ ¼ zkðgi; yN�iÞ; kX 1: ð3Þ

For k ¼ 1, (3) follows at once from (1) and 6.3 of [1]. Suppose (3) has

been proved for kW j. Set

x ¼ ð f i; yN�iÞ; z ¼ ðgi; yN�iÞ: ð4Þ

Then by the induction hypothesis,

zkðxÞ ¼ zkðzÞ; kW j: ð5Þ

Hence

gijðzjðxÞÞ ¼ gijðx1ðxÞ; x2ðxÞ; . . . ; xjðxÞÞ ðby 3:2Þ

¼ gijðx1ðxÞjW ; x2ðxÞjW ; . . . ; xjðxÞjWÞ ðby 4:7Þ

¼ f ij ððx1ðxÞjW ; gi0Þ; ðx2ðxÞjW ; gi1ðz1ðxÞÞÞ; . . . ;

ðxjðxÞjW ; gij�1ðzj�1ðxÞÞÞÞ ðby ð2ÞÞ

¼ f ij ððx1ðxÞjW ; gi0Þ; ðx2ðxÞjW ; gi1ðz1ðzÞÞÞ; . . . ;

ðxjðxÞjW ; gij�1ðzj�1ðzÞÞÞÞ ðby ð5ÞÞ: ð6Þ

Now by 6.2 and 6.5 of [1], we have

xkðzÞ ¼ sðzk�1ðzk�1ðzÞÞÞ

¼ ðuðcðzk�1ðzk�1ðzÞÞÞÞ; dðzk�1ðzk�1ðzÞÞÞÞ: ð7Þ

But by 2.7 of [1] and by 3.2,

cðzk�1ðzk�1ðzÞÞÞ ¼ zk�1ðzk�1ðzÞÞ

Applying this to (7), we obtain

xikðzÞjUik ¼ uiðzk�1ðzk�1ðzÞÞÞ

¼ ðlðzk�1ðzk�1ðzÞÞÞ; zik�1ðzk�1ðzÞÞÞ ðby 4:3Þ

¼ ðlðzk�1ðzk�1ðzÞÞÞ; gik�1ðzk�1ðzÞÞÞ ðby ð4ÞÞ:

Hence

xikðzÞjPi ¼ gik�1ðzk�1ðzÞÞ; kX 1:


Applying this to (6), we obtain

gijðzjðxÞÞ ¼ f ij ððx1ðxÞjW ; xi1ðzÞjPiÞ; . . . ; ðxjðxÞjW ; xijðzÞjPiÞÞ

¼ f ij ððx1ðxÞjW ; xi1ðxÞjPiÞ; . . . ; ðxjðxÞjW ; xijðxÞjPiÞÞ ðby ð5ÞÞ

¼ f ij ðx1ðxÞjUi1; . . . ; xjðxÞjUi

kÞ ðby 4:3Þ

¼ f ij ðx1ðxÞ; . . . xjðxÞÞ ðby 4:8Þ

¼ f ij ðzjðxÞÞ:

Applying (4), we obtain

xjðzjðxÞÞ ¼ zjðzjðxÞÞ: ð8Þ

Hence

xjþ1ðzÞjUjþ1 ¼ uðzjðzjðzÞÞÞ ðby 6:5 of ½1�Þ

¼ uðzjðzjðxÞÞÞ ðby ð5ÞÞ

¼ uðxjðzjðxÞÞÞ ðby ð8ÞÞ

¼ xjþ1ðxÞjUjþ1:

Hence by 3.2,

zjþ1ðzÞjU1 � � � � �Ujþ1 ¼ zjþ1ðxÞjU1 � � � � �Ujþ1; ð9Þ

and since

zjþ1ðzÞjR1 � � � � � Rjþ1 ¼ de � � � � � de ¼ zjþ1ðxÞjR1 � � � � � Rjþ1;

we conclude from (9) that

zjþ1ðzÞ ¼ zjþ1ðxÞ:

This completes the induction and the proof of (3). Applying 6.6 and 6.7

of [1] to (3), we obtain

Hkð f i; yN�iÞ ¼ Hkðgi; yN�iÞ; kX 1:

Hence by 4.4,

f i F gi:

But gi is by its definition a supergame p�-strategy, and thus our proof is

completed.

Parallel to Definition 3.5, we may make the following definition:

definition 4.10 A strong equilibrium p�-point f is a summable supergame

p�-strategy vector for which there is no BHN and supergame p�-strategy

vector g satisfying 3.6 and 3.7.


The set of all strong equilibrium p�-points is denoted by Sp� . If 3.7 is

replaced by 3.8, the resulting set of points is denoted by ~SSp� .

If we can succeed in restricting our considerations to supergame p�-

strategies then we will have considerably simplified our problem, because

then the information available about previous plays is the same for

all players (so that the information function may be regarded as 1-

dimensional rather than n-dimensional). That we may without loss of

generality restrict ourselves in this way is the content of the next theorem.

theorem 4.11 In a game C of perfect information, a summable super-

game p-strategy vector f is a strong equilibrium p-point if there is a strong

equilibrium p�-point f� equivalent to f.

Proof Suppose

f B Sp:

Then there is a BHN and a supergame p-strategy vector g satisfying 3.6

and 3.7. In accordance with 4.9, there is a supergame p�-strategy B-vector

gB� for which

gB� F gB: ð1Þ

Define

gN�B� ¼ f N�B

� : ð2Þ

By hypothesis we have

f N�B� F f N�B: ð3Þ

Combining (1), (2), (3), and 3.6, we obtain

g� F g: ð4Þ

From (4), 3.4 and 4.5 it follows that

SkðgÞ ¼ Skðg�Þ: ð5Þ

for each k. By hypothesis,

f F f�: ð6Þ

Applying 6.8 of [1] and 4.5 to (6), we obtain

Hð f Þ ¼ Hð f�Þ: ð7Þ

From 3.7, (5), and (7) it follows that

lim supk!y

mini A B

ðSikðg�Þ �Hið f�ÞÞ > 0: ð8Þ


From (2), (8), and 4.10 it follows that f� B Sp� , which contradicts the hy-

pothesis. This completes the proof.

corollary 4.12 HðSp� ÞHHðSpÞ.

Proof Follows from 6.8 of [1], 4.5, and 4.11.

The following theorems (4.13 through 4.16) will not be used in the

sequel; they are included for the sake of completeness. The proofs use the

same ideas as those already given, and will be omitted.

theorem 4.13 Conversely to 4.11, a summable supergame p-strategy vec-

tor f is a strong equilibrium p-point only if there is a strong equilibrium

p�-point f� equivalent to f.

corollary 4.14 HðSpÞ ¼ HðSp� Þ.

theorem 4.15 Sp� H ~SSp� .

theorem 4.16 Hð ~SSpÞ ¼ Hð ~SSp� Þ.

Theorems analogous to 4.11 and 4.13 for Sp� may also be proved.

For supergame p�-strategy vectors f, formulas 6.3 through 6.7 of [1]

may be rewritten as follows:

z1 ¼ lð f0Þ ð4:17Þ

zk ¼ ðzk�1; xkÞ ð4:18Þ

xk ¼ lð fk�1ðzk�1ÞÞ ð4:19Þ

Ek ¼ Eð fk�1ðzk�1ÞÞ ð4:20Þ

Hk ¼ HðEkÞ: ð4:21Þ

Here we are making use of the notation introduced in convention 4.7.

5 The Main Theorem

We make use of two lemmas. The first tells us that at an acceptable point

in a game of perfect information, N � B can always retaliate for a defec-

tion by B by means of a single pure strategy. The second tells us that any

payo¤ that can be obtained by a c-strategy vector in G can also be

obtained by a supergame p-strategy vector (or even by a supergame p�-

strategy vector).

lemma 5.1 Let G be a game of perfect information. Let BHN, and let h

be a vector. If there is a cN�B A CN�B such that for all cB A CB, there is an

i A B for which


HiðcB; cN�BÞW hi; ð1Þ

then there is a pN�B A PN�B such that for all cB A CB, there is an i A B for

which

HiðcB; pN�BÞW hi: ð2Þ

Proof HBðCB; cN�BÞ is easily seen to be a convex subset of the eucli-

dean B-space RB. By the hypothesis of the lemma, HBðCB; cN�BÞ cannotintersect the open ‘‘corner’’ or ‘‘octant’’ in RB given by the inequalities

xi > hi; i A B: ð3Þ

This ‘‘corner’’ is also convex. Applying the separation theorem for con-

vex sets3, we obtain a hyperplaneXi A B

aixi ¼ k ð4Þ

which passes through hB, and which separates HBðCB; cN�BÞ from the

‘‘corner’’ given by (3). In other words, we haveXi A B

aihi ¼ k; ð5Þ

and we may assume without loss of generality thatXi A B

aiHiðcB; cN�BÞW k for cB A CB ð6Þ

andXi A B

aixi > k for xB satisfying ð3Þ: ð7Þ

(if the inequalities (6) and (7) are reversed, then we may obtain them in

the given form by multiplying both sides of (4) by �1). From (3) and (7)

it follows that

aB X 0: ð8Þ

Since (4) defines a hyperplane, there must also be an i A B for which

ai 6¼ 0: ð9Þ

Define a two-person, zero-sum game G� as follows: There are two

players, 1 and 2. The game tree of G� is the same as that of G, and G� is

also a game of perfect information. Player 1 has all the moves that mem-

3. See for instance [10], pp. 29 and 81.


bers of B have in C, and player 2 has all the moves that members of

N � B have in G. Thus the mixed strategy space of player 1 is CB, and

the mixed strategy space of player 2 is CN�B (we will also use the nota-

tion M1 and M2 for these mixed strategy spaces). The payo¤ in G� will

be denoted by H�; it is defined by

H1� ðpB; pN�BÞ ¼

Xi A B

aiHiðpB; pN�BÞ ð10Þ

H2� ¼ �H1

� : ð11Þ

From (10) it follows that

H1� ðcB; cN�BÞ ¼

Xi A B

aiHiðcB; cN�BÞ ð12Þ

for all cB A CB. Combining (12) with (6) and the hypothesis of the lemma,

we obtain the existence of a cN�B A CN�B such that for all cB A CB, we have

H1� ðcB; cN�BÞW k:

Restated in terms of mixed strategies in C�, we have the existence of a

mixed strategym2� A M2

� (namely cN�B), such that for allm1� A M1

� , we have

H1� ðm1

�;m2�ÞW k: ð13Þ

By (11), G� is zero-sum as well as two-person. (13) merely tells us that

vðG�ÞW k; ð14Þ

where vðG�Þ denotes the value of G�. By the theorem of von Neumann on

two-person zero-sum games of perfect information, we have the existence

of an optimal pure strategy for player 2 in G�. Hence there is a p2� A p2�(i.e. a pN�B A PN�B) such that for all m1

� A M1� (i.e. for all cB A CB), we

have

H1� ðm1

�; p2�ÞW vðG�Þ

(i.e., by (10)),Xi A B

aiHiðcB; pN�BÞW vðG�ÞÞ: ð15Þ

Combining (5), (14), and (15), we obtainXi A B

aiHiðcB; pN�BÞWXi A B

aihi ð16Þ

for all cB A CB.


From (16) it follows thatXi A B

aiðHiðcB; pN�BÞ � hiÞW 0

for all cB A CB. Combining this with (8) and (9), we obtain for each

cB A CB, the existence of the least one i A B for which

HiðcB; pN�BÞ � hi W 0

This completes the proof of the lemma.

The next lemma tells us that the non-negative integers can be parti-

tioned into disjoint subsets whose asymptotic densities will yield an arbi-

trary finite set of non-negative real numbers adding up to 1.

lemma 5.2 Let Z be a finite set, and let y A CðZÞ. For any mapping p

from the set K of all non-negative integers into Z, and for any k A K and

z A Z, let rpðk; zÞ denote the number of j A K for which

jW k

and

pð jÞ ¼ z:

Then there is a p for which

limk!y

rpðk; zÞk þ 1

¼ yðzÞ

for all z A Z.

pð jÞ will also be denoted by pðj; yÞ, and rpðk; zÞ by rðk; z; yÞ.The proof is not di‰cult. It will be omitted.

theorem 5.3 In a game G of perfect information,

HðAcÞHHðSp� Þ:

Proof In its main outlines, the proof is analogous to that of Theorem 3

of [1], which states that HðAcÞHHðScÞ. The details, however, di¤er

considerably in the two cases. Both proofs are divided into three parts:

Given an acceptable payo¤ vector h, we must first find a sequence of

strategy vectors which will yield a payo¤ of h in the supergame (under the

assumption that the players are all ‘‘loyal’’). Next, we must find a way to

determine which players, if any, are disloyal; and finally, we must find a

way to punish the disloyal players. All these elements must be incor-

porated into a supergame strategy vector. In Theorem 3 of [1], the first


of these tasks was accomplished by having the players play the same

c-strategy vector on each play, namely the one that yields an expected

payo¤ of h. Here this cannot be done, because the players must restrict

themselves to pure strategies on each play. They must therefore play dif-

ferent pure strategy vectors on di¤erent plays in such a way so that the

limiting payo¤ is h; to show that this can be done, use must be made of

Lemma 5.2. As for the second task, this was accomplished in Theorem 3

of [1] by simply noting the make-up of the coalitions; here this cannot be

done, because in supergame p�-strategy vectors, there are no coalitions.

Instead, use must be made of the perfect information that each player

has. Finally, a group B of disloyal players could be punished in Theorem

3 of [1] by use of the c-strategy ðN � BÞ-vector cN�B provided in the def-

inition of acceptability; here only pure strategy (N � B)-vectors may be

used, so that recourse must be had to Lemma 5.1. For a more detailed

intuitive statement of the proof, see O10 of [1].

We now give the detailed proof of 5.3.

Let h A HðAcÞ, and suppose gN A Ac is such that

HðgNÞ ¼ h: ð1Þ

Then by 4.3 of [1], for each BHN there is a cN�B A CN�B, such that for

each cB A CB, there is an i A B for which

HiðcB; cN�BÞW hi:

Applying Lemma 5.1, we obtain for each BHN a pure strategy (N � B)-

vector gN�B, such that for each cB A CB, there is an i A B for which

HiðcB; gN�BÞW hi: ð2Þ

For each jX 1, let Wj be a copy of W. Wj represents the set of possible

outcomes of the jth play. Let

Qk ¼ W1 � � � � �Wk;

Qk represents the set of possible outcomes for the first k plays, and as

such is the domain of the function f ik.

Let g be any supergame p�-strategy vector in G. We define a com-

pliance function aðv1; . . . ; vk; gÞ for all ðv1; . . . ; vkÞ A Qk as follows:

definition (3) aðv1; . . . ; vk; gÞ is the maximal subset A of N for which

vj A lðgAj�1ðv1; . . . ; vj�1Þ � PN�AÞ for j ¼ 1; . . . ; k:

For each member of Qk, a tells which subset of N has been ‘‘loyal’’ to,

or has complied with, the supergame p�-strategy vector g.


It is not di‰cult to see that for each g, we have

aðzkðgÞ; gÞ ¼ N for kX 1: ð4Þ

To show (4), it is su‰cient to show that N is the maximal set satisfying

(3), i.e. that we have

xjðgÞ ¼ lðgNj�1ðzj�1ðgÞÞ � PN�NÞ; j ¼ 1; . . . ; k

¼ lðgj�1ðzj�1ðgÞÞÞ; j ¼ 1; . . . ; k:

But this follows at once from 4.19.

Moreover, it follows from (3) that

a ¼ N when k ¼ 0: ð5Þ

We are now ready to define a strong equilibrium p�-point whose payo¤

is h.

For kX 0 and qk A Qk, define

fkðqkÞ ¼ pðk; gNÞ; if aðqk; f Þ ¼ N

faðqk ; f Þk ðqkÞ ¼ gaðqk ; f Þ

fN�aðqk ; f Þk ðqkÞ ¼ arbitrary

9=; otherwise.

8>>><>>>:

ð6Þ

Definition (6) is a recursive definition; aðqk; f Þ depends only on

f0; . . . ; fk�1 not on fk.

Set zk ¼ zkð f Þ for kX 1. We first prove

fkðzkÞ ¼ pðk; gNÞ for kX 0: ð7Þ

For k > 0, (7) follows from (6) and (4); for k ¼ 0, it follows from (6) and

(5).

Combining (7) with 4.20 and 4.21, we obtain

Hkþ1ð f Þ ¼ Hðpðk; gNÞÞ for kX 0: ð8Þ

Hence

Xkþ1

r¼1

Hrð f Þ ¼Xkr¼0

Hrþ1ð f Þ

¼Xkr¼0

Hðpðr; gNÞÞ ðby ð8ÞÞ

¼Xy A P

rðk; y; gNÞHðyÞ ðby 5:2Þ:

Hence


limk!y

1

k

Xkr¼1

Hrð f Þ ¼ limk!y

1

k þ 1

Xkþ1

r¼1

Hrð f Þ

¼ limk!y

1

k þ 1

Xy A P

rðk; y; gNÞHðyÞ

¼Xy A P

HðyÞ limk!y

1

k þ 1rðk; y; gNÞ

¼Xy A P

gNðyÞHðyÞ ðby 5:2Þ

¼ HðgNÞ

¼ h ðby ð1ÞÞ:

Applying 6.8 of [1], we obtain

Hð f Þ ¼ h: ð9Þ

By 3.17, f is also summable.

It remains to prove that f is a strong equilibrium p�-point. Suppose not.

Then there is a BHN and a supergame p�-strategy vector g satisfying 3.6

and 3.7. We must then have

lemma (10) aðzkðgÞ; f Þ is monotone decreasing with k.

Proof By 4.18,

zkðgÞ ¼ ðx1ðgÞ; . . . ; xkðgÞÞ:

The result now follows from (3).

From 4.19 we obtain

xjðgÞ ¼ lðgj�1ðzj�1ðgÞÞÞ

¼ lðgN�Bj�1 ðzj�1ðgÞÞ; gBj�1ðzj�1ðgÞÞÞ

A lðgN�Bj�1 ðzj�1ðgÞÞ � PBÞ

¼ lð f N�Bj�1 ðzj�1ðgÞÞ � PBÞ ðby 3:6Þ

¼ lð f N�Bj�1 ðx1ðgÞ; . . . ; xj�1ðgÞÞ � PBÞ ðby 4:18Þ:

It now follows from (3) that

N � BH aðzkðgÞ; f Þ for kX 1: ð11Þ

Combining (11) with (5), we obtain

N � BH aðzkðgÞ; f Þ for kX 0: ð12Þ


From (10) we obtain the existence of a set BðgÞHN and a non-negative

integer k0 such that

aðzkðgÞ; f Þ ¼ N � BðgÞ for kX k0: ð13Þ

Combining (12) and (13), we obtain

BðgÞHB: ð14Þ

If BðgÞ ¼ f, then from (13), we obtain

aðzkðgÞ; f Þ ¼ N for kX k0;

whence, using (10), we deduce that

aðzkðgÞ; f Þ ¼ N for kX k0: ð15Þ

Using (3) and 4.18, we deduce from (15) that

xkðgÞ ¼ lð fk�1ðzk�1ðgÞÞÞ for kX 1: ð16Þ

From (16) and 4.17 we deduce

x1ðgÞ ¼ lðf0Þ ¼ x1ð f Þ;

and a simple inductive argument based on (16), 4.18 and 4.19 leads to the

conclusion that

zkðgÞ ¼ zkð f Þ for kX 1:

Applying 4.20 and 4.21, we obtain

HkðgÞ ¼ Hkð f Þ for kX 1: ð17Þ

From 6.8 of [1], 3.4, and (17) it follows that

limk!y

SkðgÞ ¼ Hð f Þ;

which contradicts 3.6. Thus the assumption BðgÞ ¼ f has led to a con-

tradiction, and we may conclude that

BðgÞ 6¼ f: ð18Þ

Combining (6), (13), and (18), we obtain

fN�BðgÞk ðzkðgÞÞ ¼ gN�BðgÞ for kX k0: ð19Þ

Let m be the payo¤ function defined on W, so that

H ¼ m � l: ð20Þ

Our m is what is called h in [3]; it may also be defined by

m ¼ cjW ;


where c is as in O6 of [1]. We then have

HkðgÞ ¼ HðEkðgÞÞ ðby 4:21Þ

¼ Hðgk�1ðzk�1ðgÞÞÞ ðby 4:20Þ

¼ mðlðgk�1ðzk�1ðgÞÞÞÞ ðby ð20ÞÞ

¼ mðxkðgÞÞ ðby 4:19Þ: ð21Þ

Now by (3), (13), and 4.19, we have

xkðgÞ ¼ lð f N�BðgÞk�1 ðzk�1ðgÞÞ; pBðgÞk Þ; ð22Þ

where pBðgÞr is some member of PBðgÞ.

Hence for k > k0, we have

HkðgÞ ¼ mðxkðgÞÞ ðby ð21ÞÞ

¼ ðm � lÞðf N�BðgÞk�1 ðzk�1ðgÞÞ; pBðgÞk Þ ðby ð22ÞÞ

¼ HðgN�BðgÞ; pBðgÞk Þ ðby ð20Þ and ð19ÞÞ:

Hence for k > k0, we have by the linearity of H that

1

k � k0

Xkr¼k0þ1

HrðgÞ ¼1

k � k0

Xkr¼k0þ1

HðgN�BðgÞ; pBðgÞr Þ

¼ H gN�BðgÞ;Xkþ1

r¼k0

� 1

k � k0pBðgÞr

!ð23Þ

Applying (2), we obtain the existence of an i A BðgÞ such that

Hi gN�BðgÞ;Xkþ1

r¼k0

� 1

k � k0pBðgÞr

!� hi W 0: ð24Þ

Combining (23) and (24), we deduce that

mini A BðgÞ

1

k � k0

Xkr¼k0þ1

HirðgÞ

!� hi

!W 0;

from this and (14) it follows that

mini A B

1

k � k0

Xkr¼k0þ1

HirðgÞ

!� hi

!W 0: ð25Þ

Now it follows easily from the boundedness of H that as k ! y,


1

k � k0

Xkr¼k0þ1

HirðgÞ ¼

1

k

Xkj¼1

Hij ðgÞ þO

1

k

� �

¼ SikðgÞ þ oð1Þ ðby 3:4Þ:

Applying this to (25), we obtain that as k ! y,

mini A B

ðSikðgÞ � hiÞW oð1Þ;

whence

limk!y

sup mini A B

ðSikðgÞ � hiÞW 0: ð26Þ

Applying (9), we see that (26) contradicts 3.7. This completes the proof of

5.3.


HðAcÞ ¼ HðSpÞ ¼ Hð ~SSpÞ:

In particular, h is a c-acceptable payo¤ vector in G, if and only if there is a

strong equilibrium c-point f in supergame pure strategies for which

Hð f Þ ¼ h:

Proof We have

HðAcÞHHðSp� Þ ðby 5:3Þ

HHðSpÞ ðby 4:12Þ

HHðScÞ ðby 3:13Þ

¼ HðAcÞ ðby Corollary 4 of ½1�Þ

Hence equality must hold throughout, and in particular,

HðAcÞ ¼ HðSpÞ: ð1Þ

Next, we have

HðSpÞHHð ~SSpÞ ðby 3:16Þ

HHð ~SScÞ ðby 3:12Þ

¼ HðAcÞ ðby Corollary 4 of ½1�Þ

¼ HðSpÞ ðby ð1ÞÞ:

Hence equality must hold throughout, and we deduce


HðSpÞ ¼ Hð ~SSpÞ: ð2Þ

(1) and (2) yield the first part of 5.4. The second part follows at once

from 3.11 and the first part.

corollary 5.5 Every stable4 game of perfect information has strong

equilibrium c-points in supergame pure strategies.

6 The Converse of the Main Theorem

For two-person zero-sum games not involving chance, von Neumann’s

theorem is known to ‘‘characterize’’ games of perfect information (see

[4]). More precisely, if G is a game structure of the above type which has

the property that every game that can be obtained from G (by adjunction

of a payo¤ function m) has optimal pure strategies, then G must be

equivalent to a game structure of perfect information. What can be said

in this regard for the theory presented in the previous sections?

For one thing, it is of interest to know that there are some games that

do not satisfy our main theorem (Theorem 5.4). Indeed, ‘‘matching pen-

nies’’ is such a game.

This game is given by

N ¼ ð1; 2Þ

P1 ¼ ðp11; p12Þ

P2 ¼ ðp21; p22Þ

H1ðp1i ; p2j Þ ¼1 if i ¼ j

�1 if i0 j

�H2ðpÞ ¼ �H1ðpÞ

uðpÞ ¼ p:

It is a two-person zero-sum game with value 0; hence by Theorem 1 of

[1], we have

HðAcÞ ¼ ð0; 0Þ:

If 5.4 holds for this game, then it follows that

HðSpÞ ¼ ð0; 0Þ;

and in particular, there is a summable strong equilibrium p-point f such

4. That is, every game that has any c-acceptable points (or, equivalently, any strong equi-librium c-points). See O11 of [1].


that

Hð f Þ ¼ ð0; 0Þ: ð1Þ

Define a supergame p-strategy vector g by

g2 ¼ f 2 ð2Þ

and

g1kðv1; . . . ; vkÞ ¼ p1i ; for kX 0; ðv1; . . . ; vkÞ A J1 � � � � � Jk;

where i is such that

f 2k ðv1; . . . ; vkÞ ¼ p2i :

It is then easily seen that

H1kðgÞ ¼ 1 for kX 0;

whence it follows that

S1kðgÞ ¼ 1 for kX 0:

Combining this with (1), we see that g satisfies 3.7 for B ¼ ð1Þ. By (2), g

satisfies 3.6 for B ¼ ð1Þ. Hence f cannot be a strong equilibrium p-point.

The above example constitutes a formalization of the familiar argu-

ment that states that no ‘‘scheme’’ for playing a long sequence of penny-

matchings that involves only pure strategies can be optimal.

The general statement of the converse would be as follows:

conjecture Let G be a game structure and suppose that every stable

game that is obtained from G by adjunction of a payo¤ function m has a

strong equilibrium p-point. Then G is essentially equivalent (in the sense of

[4]) to a game structure of perfect information.

There is little doubt in my mind that this conjecture is true, if not in the

given form, then at least in some other closely allied form.

7 Notation for Non-Cooperative Games

We will make use of the notion of mixed strategies. Formally, the space

Mi of mixed strategies of player i is defined to be identical with Ci. If B is

a subset of N, then we define

MB ¼Yi A B

Mi; ð7:1Þ


the cartesian product is meant. It follows that

MB HCB;

the opposite inequality is generally false. The prefix m- is an abbreviation

for ‘‘mixed.’’ The definitions relating to payo¤ remain unchanged.

8 Acceptable Points for Non-Cooperative Games

The non-cooperative game di¤ers from the cooperative game chiefly in

that the use of correlated strategy vectors that are not also mixed strategy

vectors is forbidden. The definition of acceptability for non-cooperative

games will therefore be the same as that for cooperative games (see [1],

section 4), except that correlated strategy vectors must be replaced

throughout by mixed strategy vectors. The intuitive reasoning behind the

definition remains unchanged. It might be objected that the ‘‘concerted

action’’ that is necessary to prevent a set of players B from obtaining a

payo¤ that is higher than at an acceptable point, is forbidden under non-

cooperative rules. In fact, such concerted action will probably arise any-

way as part of a ‘‘silent gentlemen’s agreement’’ among the players of

N � B. The only restriction is that though the players may ‘‘cooperate’’

in this sense (indeed, they cannot be prevented from so doing), they may

not correlate their mixed strategies before a play.

Further intuitive discussion of the notion of m-acceptability will be

found in a subsequent paper, devoted exclusively to acceptable points in

non-cooperative games.

The formal definitions are as follows:

definition 8.1 Let m0 A M. m0 is said to be m-acceptable if there is no

BHN such that for each mN�B A MN�B, there is an mB A MB for which

HBðmB;mN�BÞ > HBðm0Þ:

The set of all m-acceptable m-strategy vectors is denoted by Am. Like

c-acceptability, m-acceptability is a ‘‘global’’ notion (see [1], O4).

definition 8.2 A payo¤ vector h is said to be m-acceptable, if for some

m A Am, we have

HðmÞ ¼ h:

The following is a trivial restatement of 8.2:

theorem 8.3 A payo¤ vector h is m-acceptable if and only if for each

BHN, there is an mN�B A MN�B, such that for all mB HMB, there is an


i A B for which

HiðmB;mN�BÞW hi:

We remark that as in the cooperative case, all two-person games have

m-acceptable points. When we go beyond two-person games we find

games that have no m-acceptable points. The example given in O11 of [1]

holds for the non-cooperative case as well, as does the intuitive discussion

following the example.

We remark also that even in the two-person case, there are games of

perfect information that have no m-acceptable points in pure strategies.

See O2 of this paper, which applies unchanged in its entirety to the non-

cooperative case.

9 Equivalence ofm-Acceptability and c-Acceptability in Games of Perfect Information


HðMÞ ¼ HðCÞ

Proof HðMÞHHðCÞ follows at once from MHC. It remains to prove

HðCÞHHðMÞ:

Instead of proving this, we will prove a more general version that we will

need later. What we need for 9.1 follows from 9.2 if we set B ¼ N.

lemma 9.2 Let G be a game of perfect information. Then with each

cB A CB, we may associate an mB A MB, such that for all cN�B A CN�B we

have

HðcB; cN�BÞ ¼ HðmB; cN�BÞ:

Proof Fix cB. Because of the linearity of H, it is su‰cient to prove that

there is an mB such that for all pN�B A PN�B we have

HðcB; pN�BÞ ¼ HðmB; pN�BÞ: ð1Þ

Let b be the cardinality of B. With each i A B, we may associate an

n� bþ 1 person game Gi as follows: The players are 0 and the members

of N � B. (Intuitively, 0 represents the coalition of all the members of B.)

The set of pure strategies of 0 is PB, while the set of pure strategies for a

member j of N � B is Pj. The payo¤ to 0 is given by Ei, to members j of

N � B by E j. To avoid confusion, we will denote the payo¤ in Gi by Ei,

the expected payo¤ by Hi. Ei and Hi are ðð0ÞWN � BÞ-vectors.


From the definition of Gi, we see that for all pN�B A PN�B, we have

HN�BðcB; pN�BÞ ¼ HN�Bi ðcB; pN�BÞ

HiðcB; pN�BÞ ¼ H0i ðcB; pN�BÞ:

(ð2Þ

In Gi, cB is a mixed strategy of player 0. Let b� be its behavior (see [9],

O5, which will be called (�) in the sequel; Definition 16). Since Gi depends

on i only because of its payo¤, and since the behavior of a mixed strategy

has nothing to do with the payo¤, b� is independent of i. Since G is of

perfect information, so is Gi, and hence in particular, Gi is of perfect

recall. Noting that every pure strategy is also a behavior strategy, and in

fact its own behavior, and applying Theorem 4 of (�), we obtain that for

all pN�B A PN�B,

HiðcB; pN�BÞ ¼ Hiðb�; pN�BÞ: ð3Þ

Returning to the game G, define behavior strategies bi for each i A B by

bi ¼ b�jUi;

where Ui is the set of information sets for player i.

Then from Definitions 14 and 15 of (�) it follows that for all

pN�B A PN�B,

H0i ðb�; pN�BÞ ¼ HiðbB; pN�BÞ

HN�Bi ðb�; pN�BÞ ¼ HN�BðbB; pN�BÞ:

(ð4Þ

Combining (2), (3), and (4), we obtain that for all pN�B A PN�B,

HN�BðcN�B; pN�BÞ ¼ HN�BðbB; pN�BÞ

and for all i A B, HiðcN�B; pN�BÞ ¼ HiðbB; pN�BÞ; that is,

HðcN�B; pN�BÞ ¼ HðbB; pN�BÞ: ð5Þ

If mi is the mixed strategy corresponding to bi in accordance with

Lemma 3 of (�), then it follows from Lemma 3 and Theorem 4 of (�) that

for all pN�B A PN�B,

HðbB; pN�BÞ ¼ HðmB; pN�BÞ: ð6Þ

Combining (5) and (6), we obtain (1).

The following theorem will not be used in the sequel. It is included for

the sake of completeness.

corollary 9.3 In a game G of perfect information

EðMÞ ¼ EðCÞ:


Proof It is clear that EðMÞHEðCÞ. To prove EðCÞHEðMÞ, let c A C.

If m is the payo¤ function on W, we have

EðPÞ ¼ mðWÞ;

and indeed

E ¼ m � l: ð1Þ

Hence if

c ¼X�p A P

cðpÞp A C;

then

HðcÞ ¼Xp A P

cðpÞmðlðpÞÞ

A HðCÞHHðMÞ ðby 9:1Þ:

It follows that there is a mixed strategy vector m such that

Xp A P

cðpÞmðlðpÞÞ ¼Xp A P

Pi A N

miðpiÞ� �

mðlðpÞÞ: ð2Þ

Let us fix the coe‰cients cðpÞ, and consider a game G0 which is the

same as G except for its payo¤, which is such that the mðwÞ form a set

that is linearly independent over the field generated by the coe‰cients

cðpÞ over the rationals. For this game G0, a mixed strategy vector m may

be formed that satisfies (2). Both sides of (2) can then be considered as

linear combinations of distinct terms of the form mðwÞ, and it follows

from the way we have chosen G0 that the coe‰cients of the same terms

on both sides of (2) must be equal, i.e.,

Xp A l�1ðwÞ

cðpÞ ¼X

p A l�1ðwÞ

Yi A N

miðpiÞ !

;w A W : ð3Þ

Now (3) is seen to hold independent of the payo¤; hence no matter how m

is defined, we may write

X�w A W

Xp A l�1ðwÞ

cðpÞ

0@

1AmðwÞ ¼

X�w A W

Xp A l�1ðwÞ

Yi A N

miðpiÞ !0

@1AmðwÞ: ð4Þ

(note that the outer sum is to be considered a probability distribution

rather than an ordinary sum). From (4) we deduce


X�p A P

cðpÞmðlðpÞÞ ¼X�p A P

Yi A N

miðpiÞ !

mðlðpÞÞ;

whence, applying (1), we obtain

EðcÞ ¼X�p A P

Yi A N

miðpiÞ !

EðpÞ ¼ EðmÞ:


corollary 9.4 In a game G of perfect information,

HðAmÞHHðAcÞ:

Proof Suppose h B HðAcÞ. Then there is a BHN, such that for all

cN�B A cN�B, there is a cB A CB such that

HBðcB; cN�BÞ > hB: ð1Þ

In particular, for all mN�B A MN�B, there is a cB A CB such that

HBðcB;mN�BÞ > hB: ð2Þ

If we let mB be the mixed strategy B-vector associated with cB in accor-

dance with Lemma 9.2, then we have

HðmB;mN�BÞ ¼ HðcB;mN�BÞ: ð3Þ

Combining (2) and (3), we obtain that for each mN�B, there is an

mB A MB for which

HBðmB;mN�BÞ > hB:

Hence h B HðAmÞ, and the corollary follows.


HðAcÞHHðAmÞ:

Proof Suppose h A HðAcÞ. Then for all BHN, there is a cN�B A CN�B,

such that for all cB A CB, there is an i A B for which

HiðcB; cN�BÞW hi: ð1Þ

Let mN�B be the mixed strategy (N � B)-vector associated with cN�B

in accordance with Lemma 9.2. It then follows from 9.2 that for all

cB A CB,

HðcB;mN�BÞ ¼ HðcB; cN�BÞ; ð2Þ


and combining (1) and (2), we obtain that for all cB A CB, there is an

i A B for which

HiðcB;mN�BÞW hi: ð3Þ

In particular, for all mB A MB, there is an i A B for which

HiðmB;mN�BÞW hi;

and since this holds for all BHN, it follows that h A HðAmÞ, q.e.d.


HðAcÞ ¼ HðAmÞ:


Am ¼ Ac XM:

Proof If m A Am, then certainly

m A M: ð1Þ

But from 9.5 it follows that HðmÞ A HðAcÞ. Since among c-strategy

vectors, the property of c-acceptability is a global one, depending only

on the payo¤, it follows that

m A Ac: ð2Þ

Combining (1) and (2), we obtain m A Ac XM.

Next, let c A Ac XM. Then c A M. We also have HðcÞ A HðAmÞ, andsince among m-strategy vectors, the property of m-acceptability is a

global one, depending only on the payo¤, it follows that

c A Am:


Because of 9.6 and 9.7, we are justified in dropping the qualifying

prefix from the word ‘‘acceptable’’ when discussing games of perfect

information.

10 Supergame Strategies in the Non-Cooperative Case

A supergame strategy vector for a non-cooperative game is the same as a

supergame strategy vector for a cooperative game, except that coalitions

are forbidden. Formally, we have

definition 10.1 A supergame m-strategy f i for player i is a supergame c-

strategy for which


eð f ikðyÞÞ ¼ ðiÞ

for all kX 0 and y A Ji1 � � � � � Ji

k.

The following theorem follows at once from 10.1:

theorem 10.2 For a supergame m-strategy vector f, we have

cð fkðyÞÞ A M

for all kX 0 and y A J1 � � � � � Jk.

Parallel to the definition of strong equilibrium c-point for

cooperative games (O7 of [1]), we may make the following definition for

non-cooperative games:

definition 10.3 Let f be a summable supergame m-strategy vector. f is a

strong equilibrium m-point if there is no BHN for which there is a super-

game m-strategy vector g satisfying 7.1 and 7.2 of [1].

The set of strong equilibrium m-points will be denoted by Sm. As in [1],

it is possible to replace 7.2 of [1] by 7.3 of [1]. The set of points thus

obtained will be denoted by ~SSm.

lemma 10.4 Fp XSc HSm.

Proof Let

f A Fp XSc: ð1Þ

Since f A Fp, it follows in particular that f is a summable supergame

m-strategy vector. Suppose

f B Sm: ð2Þ

Then there is a BHN and a supergame m-strategy vector g satisfying 7.1

and 7.2 of [1]. Since every supergame m-strategy vector is also a super-

game c-strategy vector, it follows that there is a supergame c-strategy

vector g satisfying 7.1 and 7.2 of [1]. Hence

f B Sc;

contradicting (1). Hence (1) implies the falsity of (2), and our result is

proved.

lemma 10.5 Fp XSm HSp.

Proof The proof is word for word the same as that of the second part of

Theorem 3.11 (the part beginning with the word ‘‘conversely’’; the proof

is given before the proof of the first part), except that the two occurrences


of the prefix ‘‘c-’’ must be replaced by prefixes ‘‘m-’’. It is also necessary

to remember that since g is pure, it is in particular mixed.

theorem 10.6 Fp XSm ¼ Sp.

Proof We have

Sp ¼ Fp XSc ðby 3:11ÞHSm ðby 10:4Þ:

Since

Sp HFp;

it follows that

Sp HFp XSm:

Combining this with 10.5, we obtain 10.6.

theorem 10.7 Fp X ~SSm ¼ ~SSp.

Proof The proof is similar to that of 10.6.

11 The Main Theorem for Non-Cooperative Games


HðAmÞ ¼ HðSpÞ ¼ Hð ~SSpÞ:

In particular, h is an m-acceptable payo¤ vector in G, if and only if there is

a strong equilibrium m-point f in supergame pure strategies for which

Hð f Þ ¼ h:

Proof The first part follows from 5.3 and 9.6. The second part follows

from 10.6 and from the first part.

corollary 11.2 Every stable game of perfect information has strong

equilibrium m-points in supergame pure strategies.

Finally, we remark that the discussion of O6 applies unchanged to the

non-cooperative case.

References

1. R. J. Aumann, Acceptable points in general cooperative n-person games, Contributions tothe Theory of Games IV, Annals of Mathematics Study 40, Princeton University Press,1959, pp. 287–324 [Chapter 20].


2. C. Berge, Theorie generale des Jeux a n personnes, Memorial des Sciences Mathematiques138, 1957.

3. ———, Topological games with perfect information, Contributions to the Theory ofGames III, Annals of Mathematics Study 39, Princeton University Press, 1957, pp. 165–178.

4. N. Dalkey, Equivalence of information patterns and essentially determinate games, Con-tributions to the Theory of Games II, Annals of Mathematics Study 28, Princeton Uni-versity Press, 1953, pp. 217–244.

5. H. Everett, Recursive games, Contributions to the Theory of Games III, Annals ofMathematics Study 39, Princeton University Press, 1957, pp. 47–78.

6. D. Gale and F. M. Stewart, Infinite games with perfect information, Contributions to theTheory of Games II, Annals of Mathematics Study 28, Princeton University Press, 1953,pp. 245–266.

7. D. Gillette, Stochastic games with zero stop probabilities, Contributions to the Theory ofGames III, Annals of Mathematics Study 39, Princeton University Press, 1957, pp. 179–187.

8. D. B. Gillies, Some theorems on n-person games, Ph. D. Thesis, Princeton University,1953.

9. H. W. Kuhn, Extensive games and the problem of information, Contributions to theTheory of Games II, Annals of Mathematics Study 28, Princeton University Press, 1953,pp. 193–216.

10. ———, Lectures on the theory of games, issued as a report of the Logistics ResearchProject, Princeton University, 1952.

11. O. Morgenstern, and J. von Neumann, Theory of games and economic behavior, Prince-ton University Press, 1953.

12. J. F. Nash, Non-cooperative games, Ann. Math. 54 (1951), 286–295.

13. L. S. Shapley, Stochastic games, Proc. Nat. Acad. Sci. 39 (1953), 1095–1100.

14. P. Wolfe, The strict determinateness of certain infinite games, Pacific J. Math. 5 (1955),891–897.


1280-ch21 p 355

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